Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases


Allahverdiev B.

ABSTRACT AND APPLIED ANALYSIS, 2012 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume:
  • Publication Date: 2012
  • Doi Number: 10.1155/2012/473461
  • Title of Journal : ABSTRACT AND APPLIED ANALYSIS

Abstract

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space l(w)(2)(Z) (Z:= {0, +/- 1, +/- 2,...}), that is, the extensions of a minimal symmetric operator with defect index (2,2) in the Weyl-Hamburger limit-circle cases at +/-infinity). We investigate two classes of maximal dissipative operators with separated boundary conditions, called "dissipative at -infinity" and "dissipative at infinity." In each case, we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh-Weyl function of the self-adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.